= Square-root amplitude of a density operator
{title2=$W=\sqrt\rho,\quad WW^*=\rho$}
The unique positive square root of a <density operator> is a Hilbert-Schmidt operator $W$ with $\|W\|_{\mathrm{HS}}^2=\operatorname{tr}\rho=1$. It reproduces expectations by $\operatorname{tr}(A\rho)=\operatorname{tr}(W^*AW)$. In chosen orthonormal bases, vectorizing its entries gives a normalized bipartite vector $|W\rangle\!\rangle=\sum_{r,s}W_{rs}|r\rangle|s\rangle$ whose reduced <density operator> is $WW^*=\rho$. Thus the positive operator amplitude and the vector <purification> are related but distinct objects.
More generally a coefficient matrix $C$ purifies $\rho$ when $CC^*=\rho$. For square nonsingular $C$, the matrix $U=C^{-1}\sqrt\rho$ obeys $UU^*=I$ and is unitary, so $\sqrt\rho=CU$. In an eigenbasis of $\rho$, $C=\operatorname{diag}(\sqrt{p_r})U^*$; its conjugated unitary columns give the second-factor <Schmidt decomposition> vectors. This is the matrix form of <unitary freedom of purification>.
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